FR2733869A1 - MULTI-CARRIER SIGNAL, METHOD OF CONSTRUCTING SUCH SIGNAL AND METHODS OF TRANSMITTING AND RECEIVING CORRESPONDING - Google Patents

Abstract

A multicarrier signal to be transmitted to digital receivers, particularly in a non-stationary transmission channel, corresponding to the frequency multiplexing of a plurality of elementary carriers each corresponding to a series of symbols is described, wherein two consecutive symbols are separated by a symbol time tau 0. The spacing nu 0 between two adjacent carriers in the signal is equal to half the reciprocal of the symbol time tau 0 and each carrier is subjected to spectrum formatting filtering with a bandwidth which must exceed twice the intercarrier spacing nu 0, and selected so that each symbol is highly concentrated in the time and frequency domain. Methods for transmitting and receiving such a signal are also described.

Description

 Digital multicarrier signal, method of constructing such a signal and corresponding transmission and reception methods.

1. Field of the invention General field
The field of the invention is that of the transmission or broadcasting of digital data, or of analog and sampled data, intended to be received in particular by mobiles. More specifically, the invention relates to signals produced using new modulations, and the corresponding modulation and demodulation techniques.

 For many years, attempts have been made to construct modulations adapted to strongly non-stationary channels, such as transmission channels to mobiles. In such channels, the transmitted signal is affected by fading and multiple paths. The work carried out by the CCE1T in the framework of the European project EUREKA 147 (DAB: Digital Audio Broadcasting, or Digital Audio Broadcasting) showed the interest, for this type of channel, of multicarrier modulations, and in particular of OFDM (Orthogonal Frequency Division Multiplexing).

OFDM was selected as part of this European project as the basis for the standard
DAB. This technique is also envisaged as modulation for broadcasting television programs. However, there are a number of limitations (specified later) when addressing the problem of high spectral efficiency coded modulations, such as those required for digital TV applications.

JIL Possible applications
The invention has applications in a great many fields, especially when a high spectral efficiency is desired and the channel is highly nonstationary.

 A first category of applications concerns digital terrestrial broadcasting, whether it be image, sound and / or data. In particular, the invention can be applied to synchronous broadcasting, which inherently generates multipaths of long duration. It also applies advantageously to broadcasting to mobiles.

 Another category of applications concerns digital radiocommunications.

The invention can notably find applications in digital communication systems to high-speed mobiles, for example in the context of UMTS (RACE project). It can also be considered for high speed local radio networks (HIPERLAN type).

 A third category of applications is submarine transmissions.

The submarine acoustic transmission channel is highly disturbed due to the low transmission rate of the acoustic waves in the water. This leads to a large spread of the multiple paths and the Doppler spectrum. Multicarrier modulation techniques are therefore well suited to this field, and particularly the techniques that are the subject of the present invention.

2. State of the art 2.1. Theoretical remarks on the representation of the signals
Before presenting the signals according to the invention, the known signals are described below. This description is based on a general approach of multicarrier signals defined by the inventors, and new in itself. This generalization has no equivalent in the state of the art, and is not obvious to the skilled person. It must therefore be considered as part of the invention, and not as belonging to the state of the art.

We are interested in real signals (an electrical quantity for example), finite energy, and function of time. The signals can therefore be represented by real functions of L2 (R). In addition, these signals are limited bandw and their spectrum is contained in

<tb> [fc-2stc + 2] s <SEP>
<tb> fc being the "carrier frequency" of the signal. We can therefore equivalent to represent a real signal a (t) by the complex envelope sound s (t) with:
s (t) = e'9 = 'F, [a] (t) (1)
Where FA is the analytical filter.

 The signal s (t) belongs to a vector subspace (characterized by the band limitation to +) of the space of the complex functions of a real variable and of the summable band square L2 (R). This vector space can be defined in two different ways, depending on whether it is built on the body of complexes or the body of reals. For each of these spaces, we can associate a value scalar product in C or in R and construct a Hilbert space. We will call H the Hilbert space built on the body of the complexes and HR the Hilbert space built on the real body.

dans le cas de H (2) dans le cas de HR (3)
Les normes associées sont évidemment identiques dans les deux cas:

2.2. Principes généraux de l'OFDM
Les principes généraux de l'OFDM sont par exemple présentées dans le brevet français FR-86 09622 déposé le 2 juillet 1986. L'idée de base de cette technique est de transmettre des symboles codés comme des coefficients de formes d'ondes élémentaires confinées autant que possible dans le plan temps-fréquence, et pour lesquels le canal de transmission peut être considéré comme localement stationnaire. Le canal apparaît alors comme un simple canal multiplicatif caractérisé par la distribution du module des coefficients, qui suit une loi de Rice ou de Rayleigh.The corresponding scalar products are written:

in the case of H (2) in the case of HR (3)
The associated standards are obviously identical in both cases:

2.2. General principles of OFDM
The general principles of the OFDM are for example presented in the French patent FR-86 09622 filed July 2, 1986. The basic idea of this technique is to transmit coded symbols as coefficients of confined elementary waveforms as much as as possible in the time-frequency plane, and for which the transmission channel can be considered as locally stationary. The channel then appears as a simple multiplicative channel characterized by the distribution of the modulus of the coefficients, which follows a law of Rice or Rayleigh.

 The fading protection is then provided using a weighted decision code in combination with time and frequency interleaving to ensure that the symbols in the minimum code mesh are as far as possible. affected by independent fainting.

 This coding technique with interleaving in the time-frequency plane is known as COFDM. It is for example described in document [23] (see appendix 1 (to simplify the reading, most of the references of the state of the art are listed in this appendix 1. This one, as well as appendices 2 and 3 must of course be considered as integral elements of the present description)).

 There are essentially two types of known OFDM modulations. Since the names used in the literature are often ambiguous, we will introduce here more precise new names while recalling the correspondence with the existing literature. We will use the OFDM generic name, followed by a suffix specifying the type of modulation within this family.

2.3. OFDM / OAM mol2. Theoretical principles
A first category of modulations consists of a carrier multiplex
QAM (Quadrature Amplitude Modulation), or possibly in QPSK (Quadrature
Phase Shift Keying) in the particular case of binary data. We will later designate this system under the name OFDM / QAM. The carriers are all synchronized, and the carrier frequencies are spaced from the inverse of the symbol time. Although the spectra of these carriers overlap, the synchronization of the system makes it possible to guarantee the orthogonality between the symbols emitted by different carriers.

 References [1] to [7] give a good overview of the available literature on this subject.

For simplicity in writing, and according to the novel approach of the invention, the signals will be represented by their complex envelope described above. Under these conditions, the general equation of an OFDM / QAM signal is written:

Si Iti < T0/2
(6) ailleurs
xm,n(t)=ei(2#mv0@+#)x(t-n#0) avec v0#0 = 1 (7) # étant une phase quelconque, que l'on peut arbitrairement fixer à 0.La fonction x(t) est centrée, c'est-à-dire que ses moments d'ordre 1 sont nuls, soit:
#t|x(t)|2dt=#f|X(f)|2df = 0, (8)
X(f) désignant la transformée de Fourier de x(t).The coefficients am, n take complex values representing the transmitted data. The functions xm.n (t) are translated in the time-frequency space of the same prototype function x (t):

If Iti <T0 / 2
(6) elsewhere
xm, n (t) = ei (2 # mv0 @ + #) x (tn # 0) with v0 # 0 = 1 (7) # being any phase, which can be arbitrarily fixed at 0.The function x (t) is centered, that is to say that its moments of order 1 are zero, namely:
#t | x (t) | 2dt = # f | X (f) | 2df = 0, (8)
X (f) designating the Fourier transform of x (t).

Under these conditions, we observe that: ItIXm, n (t) 2dt =
(9)
#f | Xm, n (f) | df = mv0
The barycenters of the basic functions thus form a network of the time-frequency plane generated by the vectors (T0, 0) and (0, v0), as is illustrated in FIG. 1.

This network is of unit density, ie v0T0 = 1. We can refer to article [9] for a more detailed discussion on this subject.

The prototype function x (t) has the particularity that the functions {xn, n} are orthogonal to each other, and more precisely constitute a Hilbert base of L2 (R), namely:

<tb><SEP> 1 <SEP> if (m, n) <SEP> = <SEP> (m ', n')
<tb><xm, n | xm ', n'> = <SEP>#<SEP> (10)
<tb><SEP> 0 <SEP> otherwise
<Tb>
To project a signal under this base is simply to split the signal into sequences of duration 0 and to represent each of these sequences by the corresponding Fourier series development. This type of decomposition constitutes a first step towards a location at the same time and in frequency, as opposed to the classical Fourier analysis, which ensures a perfect frequency localization with a total loss of the temporal information.

 Unfortunately, if the temporal location is excellent, the frequency localization is much worse, because of the decreasing rate of X (f). The Balian-Low-Coifman-Semmes theorem (see [9], p. 976) shows that if we call X the Fourier transform of x, tx (t) and fX (f) can not be simultaneously summable square.

2.3.2. OFDM / OAM with guard interval
In general, one can characterize the tolerance of an OFDM modulation with respect to multipath and Doppler spreading by a parameter measuring globally the variation of the intersymbol interference level (IES). function of an offset in time or frequency. The justification for this concept is given in Appendix 2. This tolerance parameter is called # and is defined by the relation:
5 1 / 4XtAf (11) with:

 By virtue of the Heisenberg inequality, 5 can not exceed unity.

Given the Balian-Low-Coifman-Semmes theorem cited previously, parameter 5 is 0 for OFDM / QAM. This is an important defect of modulation
OFDM / QAM as described above. This is characterized in practice by a high sensitivity to temporal errors, and therefore to multiple paths.

 This defect can be circumvented by the use of a guard interval described for example in [5]. This is an artifice of extending the rectangular window of the prototype function. The density of the network of basic symbols is then strictly less than unity.

 This technique is possible because there is an infinite number of translated versions of the initial symbol within a symbol extended by a guard interval.

Of course, this only works because the prototype function is a rectangular window. In this sense, the OFDM (QAM with guard interval constitutes a single singular point.

OFDM / QAM modulation with guard interval is the basis of the system
DAB. This guard interval makes it possible to limit the interference between symbols, at the cost of a loss of performance, since part of the information transmitted is not actually used by the receiver, but only serves to absorb the multiple paths.

 Thus, in the case of the DAB system, where the guard interval represents 25% of the useful symbol, the loss is 1 dB. In addition, there is an additional loss, due to the fact that in order to obtain a given overall spectral efficiency, the loss due to the guard interval must be compensated for by a better efficiency of the code used.

 This loss is marginal in the case of the DAB system, because the spectral efficiency is low. On the other hand, if one aims at an overall spectral efficiency of 4 bits / Hz, it is necessary to use a code with 5 bits / Hz, according to the theorem of Shannon a loss of the order of 3 dB. The overall loss is in this case about 4 dB.

2.3.3. Other OFDM / OAM systems
Other OFDM / QAM systems can be imagined. Unfortunately, no filtered QAM modulation (ie using conventional half-Nyquist formatting (or, more precisely, square root of Nyquist)), checks for the required orthogonality constraints. Known prototype functions verifying the required orthogonality criteria are:
- the rectangular window;
- the cardinal sinus.

 These two examples are trivial, and appear dual to each other by Fourier transform. The case of the rectangular window corresponds to the OFDM / QAM without guard interval. The case of the cardinal sinus corresponds to a conventional frequency multiplex (that is to say whose carriers have disjoint spectra) with a roll-off of 0%, which constitutes an asymptotic case difficult to achieve in practice.

 In each of these cases, we observe that the prototype function is perfectly limited, either in time or in frequency, but has a mediocre decrease (in 1 / t or 1 / f) in the dual domain.

 The Balian-Low-Coifman-Semmes theorem leaves little hope that satisfactory solutions can exist. As indicated previously, this theorem proves that tx (t) and fX (f) can not simultaneously be summable squares. We can not therefore hope to find a function x (t) such that x (t) and X (f) decrease simultaneously with an exponent lower than -3/2.

 This does not preclude the existence of satisfactory functions in the eyes of an engineer. Nevertheless, a recent article [10] dealing with this subject exhibits another example of a prototype function having the required properties. The appearance of the prototype function proposed in this article is far removed from what one may wish in terms of temporal concentration. It is therefore probable that there is no satisfactory solution of OFDM / QAM type.

 In conclusion, the OFDM / QAM, corresponding to the use of a density network 1 and complex coefficients am.nn can be put into practice only in the case of a rectangular time window and use a guard interval. Those skilled in the art seeking other modulations are therefore led to turn to the techniques described below under the name of OFDM / OQAM.

2.4. OFDM / OOAM
A second category of modulations uses a carrier multiplex
OQAM (Offset Quadrature Amplitude Modulation). We will later designate this system under the name OFDM / OQAM. The carriers are all synchronized, and the carrier frequencies are spaced half the inverse of the symbol time. Although the spectra of these carriers overlap, the synchronization of the system and the choice of the phases of the carriers makes it possible to guarantee the orthogonality between the symbols emitted by different carriers. References [11-18] give a good overview of the available literature on this topic.

For simplicity in writing, the signals will be represented in their analytic form. Under these conditions, the general equation of an OFDM / OQAM signal is written:

 The coefficients am, n take real values representing the transmitted data.

si m + n est pair
(15) si m + n est impair avec v0#0=1/2, # étant une phase quelconque que l'on peut arbitrairement fixée égale à 0.The functions xm, n (t) are translated in the time-frequency space of the same prototype function x (t):

if m + n is even
(15) if m + n is odd with v0 # 0 = 1/2, # being any phase that can be arbitrarily set equal to 0.

 The barycenters of the basic functions thus form a network of the time-frequency plane generated by the vectors (X0, 0) and (0, v0), as illustrated in FIG.

This network is of density 2. The functions xn, .n (t) are orthogonal in the sense of the scalar product in R. In the known approaches, the prototype function is bounded in frequency, so that the spectrum of each carrier does not cover than that of adjacent carriers. In practice, the prototype functions considered are even (or possibly complex) even functions verifying the following relation:

Si If < v0 (17)
ailleurs
Lorsqu'on observe x(t) et sa transformée de Fourier, on note que X(f) est à support borné et que x(t) décroît en t-2, c'est à dire un résultat notablement meilleur que la limite théorique découlant du théorème de Balian-Low-Coifman-Semmes.Les formes d'onde élémentaires sont mieux localisées dans le plan temps fréquence que dans le cas de I'OFDM/QAM, ce qui confere à cette modulation un meilleur comportement en présence de trajets multiples et de Doppler. Comme précédemment, on peut définir le paramètre 5 mesurant la tolérance de la modulation au délai et au Doppler. Ce paramètre 5 vaut 0.865.<tb><SEP> X (f) <SEP> = <SEP> 0 <SEP> if | f | # v0
<tb> (16)
<tb> | X (f) | 2+ | X (f <SEP> - <SEP> v0) | 2 <SEP> = <SEP> 1 / v0 <SEP> if <SEP> 0 # f # v0
<Tb>
A possible choice for x (t) is the impulse response of a 100% roll-off half-Nyquist filter.

If If <v0 (17)
elsewhere
When x (t) and its Fourier transform are observed, we note that X (f) has a bounded support and that x (t) decreases at t-2, that is to say a result that is significantly better than the theoretical limit. deriving from the Balian-Low-Coifman-Semmes theorem.The elementary waveforms are better localized in the time-frequency plane than in the case of the OFDM / QAM, which confers on this modulation a better behavior in the presence of paths. multiple and Doppler. As before, parameter 5 measuring the tolerance of modulation to delay and Doppler can be defined. This parameter 5 is 0.865.

3. Disadvantages of known systems
These known systems have many disadvantages and limitations, especially in highly disturbed channels, and when high efficiency is required.

3. 1. OFDM / QAM
The main problem of the OFDM / QAM system is that it imperatively requires the use of a guard interval. As indicated above, this causes a noticeable loss of efficiency when high spectral efficiencies are targeted.

 In addition, transmitted symbols are poorly concentrated in the frequency domain, which also limits performance in highly non-stationary channels. In particular, this spread makes it difficult to use equalizers.

3.2. OFDM / OOAM
On the other hand, the frequency performance of the OFDMtOQAM is rather satisfactory and the problem of the loss related to the guard interval does not arise. On the other hand, the impulse response of the prototype function has a relatively slow temporal decay, ie in l / x2.

 This involves two types of difficulties. First of all, the waveform can hardly be truncated over a short time interval, which implies complex processing at the receiver. In addition, this also complicates possible equalization systems.

 In other words, the efficiency of the OFDM / OQAM techniques is greater than that of the OFDM / QAM, but these techniques prove more complex to implement, and therefore expensive, in particular in the receivers.

4. Presentation of the invention 4.1 Obectives of the invention
The invention particularly aims to overcome these various disadvantages and limitations of the state of the art.

 Thus, an object of the invention is to provide a digital signal intended to be transmitted or broadcast to receivers, which makes it possible to obtain better performances in non-stationary channels, and particularly in strongly non-stationary channels.

 The invention also aims to provide such a signal, to obtain a high spectral efficiency.

 Another object of the invention is to provide such a signal, which avoids the drawbacks of the OFDM / QAM related to the guard interval, while maintaining a temporal response of the prototype function as concentrated as possible, in particular to simplify the processing at the receiver.

 The invention also aims to provide such a signal, allowing the production of receivers with limited complexity and cost, in particular with regard to demodulation and equalization.

 A complementary object of the invention is to provide transmitters, receivers, transmission or broadcasting methods, reception methods and methods of construction, that is to say of definition, of a modulation corresponding to such a signal.

4.2 Main features of the invention
These objectives, as well as others which will appear later, are achieved according to the invention by a multicarrier signal intended to be transmitted to digital receivers, in particular in a non-stationary transmission channel, corresponding to the multiplexing frequency of several elementary carriers each corresponding to a series of symbols, two consecutive symbols being separated by a symbol time T0, in which, on the one hand, the spacing v0 between two neighboring carriers is equal to half of the inverse time symbol 0, and in which, on the other hand, each carrier undergoes a shaping filtering of its spectrum having a bandwidth strictly greater than twice said spacing between carriers v0.This spectrum is chosen so that each element of symbol is concentrated as much as possible in both the time domain and the frequency domain.

où :
am,n est un coefficent réel représentatif du signal source, choisi dans un alphabet
de modulation prédéterminé; m est un entier représentant la dimension fréquentielle; n est un entier représentant la dimension temporelle; t représente le temps; X,,(t) est une fonction de base, translatée dans l'espace temps-fréquence d'une
même fonction prototype x(t) paire prenant des valeurs réelles ou complexes, soit

In particular, such a signal can respond to the following equation:

or :
am, n is a real coefficent representative of the source signal, chosen from an alphabet
predetermined modulation; m is an integer representing the frequency dimension; n is an integer representing the time dimension; t represents the time; X ,, (t) is a basic function, translated in the time-frequency space of a
same prototype function x (t) pair taking real or complex values, either

where (p is an arbitrary phase parameter,
the Fourier transform X (f) of the function x (t) having a support extending to
beyond the interval [-v0, v0],
and where said basic functions {xn} are orthogonal to each other, the real part
the scalar product of two different basic functions being zero.

 The symbol "+" indicates that xn, n (t) can take either a negative or a positive sign. It does not mean, of course, that x, l, n (t) takes both values.

 Thus, the invention is based on a modulation system that uses prototype functions as concentrated as possible in the time-frequency plane. The advantage of this approach is to have modulation producing a signal avoiding the drawbacks of the OFDM / QAM related to the guard interval, while keeping a temporal response of the prototype function as concentrated as possible, so to simplify the processing at the receiver.

 In other words, the object of the invention is new modulation systems constructed as OFDM / OQAM on an orthogonal network of density 2, without the prototype function having a frequency-bound support. Among the proposed modulations, we find either modulations using prototypical functions with bounded time support, or prototypical functions which are not limited in time or frequency, but which have by counting properties of rapid decay at a time and in frequency, and a near-optimal concentration in the time-frequency plane.

 Such signals are by no means obvious to those skilled in the art, in view of the state of the art. As indicated above, there are basically two modes of construction of OFDM modulations.

The first known construction method uses a density network 1. This first solution uses a signal decomposition base where any signal is divided into intervals, each interval then being decomposed into a series of signals.
Fourier. This is the OFDM / QAM solution. The literature gives few examples of alternative solutions built on the same network, and the results obtained are of little practical interest [10].

 In addition, the OFDM / QAM technique is the only one that can benefit from the guard interval method. The OFDM / QAM solution is therefore a singular point that does not allow extensions.

The second known construction method (OFDM / OQAM) uses a density network 2. The orthogonality between symbols centered on the same frequency or on adjacent frequencies is guaranteed by a shaping of the half-type signals.
Nyquist and by the appropriate choice of the phase of the signals. Finally, the orthogonality beyond the adjacent frequencies is guaranteed by the fact that the frequency carriers are disjoint.

 Therefore, the construction of new modulations not verifying this property is not obvious.

 All the variants of the invention described below have the advantage of using a prototype function, either limited in the time domain or fast decreasing, so that the function can be easily truncated.

According to a first variant, said prototype function x (t) is an even function, zero outside the interval [- # 0, # 0], and satisfying the relation:

si |t|##0
ailleurs
Dans ce premier cas (appelé par la suite OFDM/MSK), les performances en terme de résistance au Doppler et aux trajets multiples sont équivalentes à 1'OFDM/OQAM, et la réalisation du récepteur est simplifiée.<tb><SEP> x (t) = 0 <SEP> if | t | ## 0
<tb> Ix (t) 12 <SEP><SEP> + Ix (t-r0) 12 <SEP> = <SEP> 1 / TO <SEP> if <SEP> O <SEP><<SEP> t <SEP ><
<Tb>
Advantageously, said prototype function x (t) is defined by:

if | t | ## 0
elsewhere
In this first case (hereinafter called OFDM / MSK), the performances in terms of Doppler resistance and multipath are equivalent to the OFDM / OQAM, and the realization of the receiver is simplified.

la fonction y(t) étant définie par sa transformée de Fourier Y(f):

où G(f) est une fonction gaussienne normalisée du type: G(f) = (2a)"4e2 a étant un paramètre réel strictement positif. k varie de -oo à +oo. According to a second variant of the invention, said prototype function x (t) is characterized by the equation:

the function y (t) being defined by its Fourier transform Y (f):

where G (f) is a normalized Gaussian function of the type: G (f) = (2a) "4e2 a being a strictly positive real parameter k varies from -oo to + oo.

 Advantageously, the parameter a is equal to unity. The corresponding modulation is subsequently called OFDM / IOTA. In this case, the corresponding prototype function, denoted 3, is identical to its Fourier transform.

 The realization of the receiver is simpler than in the case of the OFDM / OQAM, although slightly more complex than in the previous case, but the performances are appreciably higher.

The invention also relates to a method for transmitting a digital signal, in particular in a non-stationary transmission channel, comprising the following steps:
- channel coding of a digital signal to be transmitted, delivering coefficients
actual numerals a ", n chosen from a predetermined alphabet;
constructing a signal s (t) corresponding to the equation defined above;
transmitting a signal having for complex envelope said signal s (t) towards
least one receiver.

 Advantageously, such a method further comprises a frequency and / or time interleaving step, applied to the bits forming said digital signal to be transmitted or to the digital coefficients a ,,.

 This ensures optimal performance in nonstationary channels.

 The invention also relates to the transmitters of such a signal.

The invention also relates to a method for receiving a signal as described above, which comprises the following steps:
receiving a signal having a complex signal envelope r (t)
corresponding to the signal s (t) on transmission;
- estimation of the response of the transmission channel, including an estimate of
the phase response g, n and the amplitude response pmn;
demodulating said signal r (t), comprising the following steps:
multiplication of said signal r (t) by the prototype function x (t)
- folding of the filtered waveform modulo 2 # 0;
- application of a Fourier transform (FFr),
- correction of the phase # m, n induced by the transmission channel;
correction of the phase corresponding to the term im + n;
selection of the real part of the coeeficient obtained am, n corresponding to
coefficient am, n emitted weighted by the amplitude response # m, n of the channel of
transmission.

Preferably, this reception method comprises a de-interlacing step in frequency and / or in time of said real numerical coefficients # m, n and, possibly, corresponding values # m, n of the amplitude response of the
channel, said deinterlacing being the opposite of an interleaving implemented on transmission,
and / or a weighted decision decoding step adapted to the channel coding implemented at
transmission.

 The invention also relates to the corresponding receivers.

la fonction y(t) étant définie par sa transformée de Fourier Y(f):

Finally, the invention also relates to a preferred method of construction
of a prototype function x (t) for a signal as described above, including the
following steps:
selection of a Gaussian function G (f) normalized of the type: G (f) = (2a) H4 e ~ t
determining said prototype function x (t) such that:

the function y (t) being defined by its Fourier transform Y (f):

 This method makes it possible in particular to define the prototype function S, described above.

5. Description of particular embodiments of the invention
5.1. List of Figures
FIG. 1 illustrates a network of density 1, corresponding to that set
works in the case of the known modulation OFDM / QAM;
FIG. 2 illustrates a network of density 2, corresponding to that set
in the case of the known OFDM / OQAM modulation and in the case of
of the invention;
FIGS. 3A to 3D, 4C to 4D, 5A to 5D, 6A to 6D and 7A to 7D illustrate
respectively the known modulations OFDM / QAM (3), OFDM / QAM
with guard interval (4), OFDM / OQAM (5) and the modulations of
the invention OFDM / MSK (6) and OFDM / IOTA (7), according to the following aspects
. A: the prototype function x (t);
. B: the Fourier transform in linear of the prototype function;
.C: the module of the ambiguity function in linear (such as
defined in Annex 2);
. D: the intersymbol function (as defined in Appendix 2);
FIG. 7E shows in logarithmic scale the decay of the signal
OFDM / IOTA;
- Figure 8 the ambiguity function of a Gaussian function;
FIG. 9 is a block diagram of a transmitter (and of the method
corresponding emission) usable according to the invention;
FIG. 10 is a block diagram of a receiver (and the method of
corresponding reception) usable according to the invention
FIG. 11 illustrates more precisely the demodulation method implemented
in the receiver of Figure 10.

Theoretical principles of the signals according to the invention
All OFDM / OQAM basic functions defined in (15) can be rewritten as:

 The barycenters of the basic functions thus form a network of the time-frequency plane generated by the vectors (r0, 0) and (0, v0) (see FIG. 2). This network is of density 2, that is to say that v0 # 0 = 1/2. As indicated in [16], these functions constitute a Hilbert database of HR. In order to simplify the writing, we will omit thereafter the inversions of sign.

 In a general way, one looks for the conditions on x (t) so that the family {xm.n} constitutes a hilbertian basis of HR. We will impose that x (t) be an even function.

soit, en posant t'= t-(n+(n+n')#0/2 et #'0=(n-n')#0:

The scalar product of xm, n and of xi ni can be written:

either, by putting t '= t- (n + (n + n') # 0/2 and # '0 = (n-n') # 0:

= #e [i (m-m ') + (n-n') i (m-m ') (n + n') J (function pair + i odd function)]
Orthogonality is thus obtained if the coefficient of the integral is a pure imaginary number. The analysis of this coefficient shows that it is enough for that mm 'or that n' is odd.

soient orthogonales entre elles au sens du produit scalaire dans R. D'ailleurs, si tel est le cas, ces fonctions sont aussi orthogonales au sens du produit scalaire dans C, pour des raisons de symétrie analogues à celles évoquées ci-dessus. Une autre façon d'exprimer cette condition est d'utiliser la fonction d'ambiguïté de x [19]:

n suffit alors de trouver une fonction x(t) paire telle que:
A (2nlO, 2mv0) = O ,#(m,n)# (0,0) (24)
Si l'on compare le problème ainsi posé à celui de trouver une base hilbertienne au sens du produit scalaire dans C, les contraintes d'orthogonalité sont nettement moins fortes, puisque le réseau concerné est deux fois moins dense. En effet, les fonctions de base sont centrées sur les points du réseau {2mv0,2n#0}, c'est à dire un réseau de densité 1/2. On voit donc apparaître ici de façon intuitivement évidente les raisons de l'inapplicabilité du théorème de Balian-Low-Coifman-Semmes.The network can therefore be broken down into four sub-networks, as it appears in FIG. 2 ({m pair, n even}, {m pair, n odd odd n even}, {odd m, n odd}) which are orthogonal to each other (any function of one of the subnets is orthogonal to any function of another subnetwork). A sufficient condition for {xm, n} to be a Hilbert base is that: <xm, n | xm ', n'> R = 0 # mm 'even, # n-nair, (m, n) # (m ', n') (21)
I1 just need to find a function x (t) pair such that the functions of the type:

they are orthogonal to each other in the sense of the scalar product in R. Moreover, if this is the case, these functions are also orthogonal in the sense of the scalar product in C, for reasons of symmetry similar to those mentioned above. Another way to express this condition is to use the ambiguity function of x [19]:

It is enough then to find a function x (t) pair such that:
A (2n10, 2mv0) = O, # (m, n) # (0,0) (24)
If we compare the problem thus posed to that of finding a Hilbert basis in the sense of the scalar product in C, the orthogonality constraints are much less strong, since the network concerned is twice as dense. Indeed, the basic functions are centered on the points of the network {2mv0,2n # 0}, that is to say a network of density 1/2. The reasons for the inapplicability of the Balian-Low-Coifman-Semmes theorem are thus intuitively evident.

 In the case of OFDM / OQAM, the orthogonality of the functions X2m.2n (t) between them is obtained by two constraints of different natures. Indeed, if m # m ', <x2m, n | x2m .2n'> is null because these functions have disjoint spectra.

On the other hand, (x2m 2n | x2m 2 ") is zero because X (f) has a half-type formatting
Nyquist.

 As shown by the abundant literature already cited, the skilled person considers it imperative to verify these two constraints. In particular, he considers that the prototype function must be frequency-bound.

5.3. General principles of the invention
The invention is based on an entirely new approach of OFDM / OQAM type multicarrier signals, according to which the orthogonality is obtained no longer by the respect of the two constraints mentioned above, but by a specific definition of the prototype functions.

 In other words, the subject of the invention is new signals, based on modulation systems constructed as OFDM / OQAM on an orthogonal network of density 2, without the prototype function having a frequency-bound support. .

 The principle used is to build orthogonal networks of density 1/2, then to deduce networks of density 2 by a judicious choice of the phases of the signals.

 Very many signals can be constructed according to the technique of the invention. Two non-limiting examples of such signals, called OFDM / MSK and OFDM / IOTA respectively, are given below. A particular method for constructing such signals is also given, by way of non-limiting example, in Appendix 3. This method is of course part of the invention, and has only been included in the appendix to simplify the reading of the this description.

Si t < tu (25) ailleurs
En fait, on constate a posteriori que cette modulation peut être considérée comme duale de l'OFDM/OQAM, puisqu'elle correspond à un échange des axes temps et fréquence. L'intérêt essentiel de cette modulation par rapport à l'OFDM'OQAM est que la fonction prototype est strictement limitée dans le temps, ce qui simplifie notablement l'implémentation du récepteur, puisque le nombre de coefficients du filtre d'entrée est considérablement réduit.Par ailleurs, les performances en présence de trajets multiples sont inchangées, le paramètre 5 étant identique.5.4. OFDM / MSK modulation
We consider here a new modulation built according to the same generic equation as the OFDM / OQAM modulation (equations 14 and 15), but from a different prototype function. We will call it OFDM / MSK because each carrier is modulated in MSK [20]. The prototype function is written:

If t <you (25) elsewhere
In fact, we note a posteriori that this modulation can be considered as dual OFDM / OQAM, since it corresponds to an exchange of time and frequency axes. The essential interest of this modulation with respect to OFDM'OQAM is that the prototype function is strictly limited in time, which considerably simplifies the implementation of the receiver, since the number of input filter coefficients is considerably reduced. In addition, the performances in the presence of multiple paths are unchanged, the parameter 5 being identical.

5.5. IOTA modulation
On the other hand, OFDM / IOTA modulation results from a totally new and original approach in the field of signal processing, which we call transformed IOTA (for "Isotropic Orthogonal Transform Algorithm"), and described in appendix 3.

désignant la fonction IOTA définie en annexe 3.5.5.1. Signal equation
We consider here a new modulation built according to the same generic equation as the OFDM / OQAM modulation (equations 14 and 15), but from a different prototype function. We will call it OFDM / IOTA because of the choice of the prototype function. The prototype function is written:

designating the IOTA function defined in Annex 3.

It will be noted that the construction method given in appendix 3 makes it possible to obtain an infinity of solutions, the IOTA function constituting a remarkable solution. The basic functions of the OFDM / IOTA modulation are thus written:

avec:

5.5.2. Commentaires des figures et avantages liés à la décroissance rapide
Afin de mieux mettre en évidence, de façon visuelle, les avantages de l'invention, on présente pour chaque modulation discutée précédemment:
.A la fonction prototype x(t);
. B la transformée de Fourier en linéaire de la fonction prototype;
. C le module de la fonction d'ambiguïté en linéaire (telle que définie
dans l'annexe 2);
.D la fonction d'intersymbole (telle que définie dans l'annexe 2).The transmitted signal is written thus:

with:

5.5.2. Comments on the figures and benefits of rapid decay
In order to better highlight, visually, the advantages of the invention, it is presented for each modulation discussed above:
.A the prototype function x (t);
. B the Fourier transform in linear of the prototype function;
. C the linear ambiguity function module (as defined
in Annex 2);
.D the intersymbol function (as defined in Annex 2).

The views of the ambiguity function (subscripted figures C) make it possible to judge the confinement in the time-frequency plane of the prototype function. The views of the intersymbol function (subscripted figures D) make it possible to appreciate the sensitivity of a modulation to the delay and the Doppler. Phase errors are not considered, since
all modulations are equivalent in this respect.

 FIGS. 3A to 3D relate to the known case of conventional OFDM / QAM. The main defect of this modulation is not, as the frequency response of the prototype function might suggest, the slight decrease in the level of the sidelobes.

 In fact, the sensitivity of the OFDM to frequency errors is only slightly higher than that of the other modulations considered. The IES, on the other hand, has a different statistic, which results in a horizontal closure of the eye equivalent to that of a zero roll-off modulation. There are therefore traces, certainly unlikely, but which can create systematic errors in the absence of coding. This detail is "unsightly", but without real consequences in the presence of coding. On the other hand, this weak decrease makes that the energy of IES is distributed on a large number of neighboring symbols, which makes any attempt of equalization very delicate.

Paradoxically, the real problem comes from the sudden limitation of the temporal response, which corresponds to a triangular ambiguity function along this axis. This gives an intersymbol function with a very high sensitivity to temporal errors
the slope is vertical and parameter 5 is zero. This justifies the use of a guard interval.

 Figures 4C and 4D relate to the OFDM / QAM with a guard interval (the prototype and Fourier transform functions are identical to those of the OFDM / QAM illustrated in Figures 3A and 3C. creates a flat area at the level of the ambiguity function.In fact, we should rather speak in this case of crossambiguity.There is obviously this flat part at the intersymbol function, which gives immunity to temporal errors. The figures represent the case of a guard interval 0.25 T0.

 In terms of frequency errors, the properties are the same as those of the standard OFDM.

 The cost of the guard interval is allowable when considering low spectral efficiency modulations. On the other hand, n becomes prohibitive if one seeks a high spectral efficiency: let us take for example a guard interval equal to a quarter of the useful symbol. Under these conditions, it is necessary to achieve a net efficiency of 4 bits / s / Hertz a modulation and coding system having a raw efficiency of S bits / s / Hertz, a loss of 3 dB compared to the limit capacity of Shannon. It is also necessary to add to this loss the additional loss of 1 dB due to the power "unnecessarily" emitted in the guard interval. In total, 4 dB are lost compared to the optimum.

 Figures SA to SD present the case of OFDM / OQAM.

 The time response of OFDM / OQAM is better than that of OFDM / QAM. Nevertheless the temporal decay is only in l / t2. The ambiguity function is canceled on a 1/2 density network. The sensitivity to frequency errors is greater than that to temporal errors. Parameter 5 is 0.8765.

FIGS. 6A to 6D relate to the first embodiment of the invention,
I'OFDM / MSK. It is verified that it has properties that are strictly identical to those of the OQAM by reversing the time and frequency scales. Parameter 5 is unchanged.

 Finally, FIGS. 7A to 7D show the OFDM / IOTA modulation. This has a rapid decrease (in the mathematical sense of the term) in time and frequency, which allows to consider the equalization in the best possible conditions.

It also has perfect symmetry with respect to these two axes. Its intersymbol function is almost ideal. In general, his behavior is similar to that of Gaussian. Parameter 5 is 0.9769.

 The function of ambiguity of the function S (FIG. 7C) can be compared to that of a Gaussian, as illustrated in FIG. 8. The general appearance of these two functions is very similar at the level of the vertex. They differ on the other hand at the base.

FIG. 7E shows in logarithmic scale the decay in signal time
IOTA. It is observed that the amplitude of the signal decreases linearly in logarithmic scale (in time and in frequency, of course, since the two aspects are identical), or exponentially on a linear scale. This property therefore makes it possible in a practical embodiment to truncate the waveform and thus limit the complexity of the receiver.

5.6. Principle of a transmitter
FIG. 9 presents a simplified block diagram of a transmitter of a signal according to the invention. The emission process is deduced directly.

We consider a high bit rate binary source (typically a few dozen
Megabits / s). By binary source is meant a series of data elements corresponding to one or more source signals of all types (sounds, images, data) sampled digital or analog. These binary data are subjected to a binary-to-binary channel coding adapted to fading channels. For example, it would be possible to use a trellis coded modulation code, possibly concatenated with a code of
Reed-Solomon. More precisely, if it is desired to have a spectral efficiency of 4 bits / hertz, a 2/3 efficiency code associated with an 8AM modulation can be used, taking 8 amplitude levels.

 Then, in accordance with the principle set forth in patent FR-88 15216, these coded data are distributed (93) in the time-frequency space so as to provide the necessary diversity, and to decorrelate the Rayleigh fading (fading) affecting the emitted symbols.

 More generally, a first binary to binary coding, a time and frequency interleaving and a binary coding with coefficients, commonly called mapping, are carried out. It is clear that the interleaving can be indifferently carried out before or after the mapping, according to the needs and the codes used.

 At the end of this coding operation, the actual symbols to be transmitted amn are available. The embodiment of the OFDM / MSK or OFDM / IOTA modulator 94 is analogous to that of an OFDM / OQAM transmitter. Only the prototype waveform differs.

We can refer to [15] for a detailed description of the modulation system. To construct the signal to be emitted, we group together the symbols of the same rank n, and we calculate:

 This operation may advantageously be performed in digital form by a fast Fourier transform (for) 1) covering all the symbols of the same rank n, followed by a multiplication of the resulting waveform by the prototype function IOTA, and finally an addition of the symbols of different ranks (summation according to the index n).

 The complex signal thus generated is then converted into analog form 98, then transposed to the final frequency by a modulator 99 with two channels in quadrature (modulator I &), and finally amplified 910 before being transmitted 911.

5.7. Principle of a receiver
Figure 10 schematically illustrates a receiver of a signal according to the invention (as well as the corresponding reception method).

 The OFDM / MSK or OFDM / IOTA receiver is substantially analogous to that adapted to OFDM / OQAM modulation. Entrance floors are conventional. The signal is preamplified 101, then converted to intermediate frequency 102 in order to carry out channel filtering 103. The intermediate frequency signal is then converted into baseband on two quadrature channels 105. In addition, the automatic correction functions are performed. gain (AGC) 104, which controls preamplification 101.

 Another solution is to transpose the intermediate frequency signal on a low carrier frequency, so as to sample the signal on a single channel, the complex representation is then obtained by digital filtering. Alternatively, the RF signal can be transposed directly to the baseband (direct conversion), the channel filtering being then performed on each of the two channels I &. In all cases, it can be reduced to a discrete representation of the signal of the complex envelope corresponding to the received signal.

In order to detail the digital processing in baseband, we will consider a multicarrier type modulation characterized by the equation of the complex envelope of the emitted signal:

Either a transmission channel characterized by its variable transfer function T (f, t) (see annex 2). The complex envelope of the received signal r (t) is written:

The demodulator estimates (106) the transfer function T (f, t) by conventional means, which can for example use an explicit carrier reference network according to patent FR-90 01491. To demodulate the signal itself (107) , we assimilate the channel locally to a multiplicative channel characterized by an amplitude and a phase corresponding to the value of T (f, t) for the moment and the frequency considered. To estimate am, n (t), the received signal is therefore assimilated to the signal:
r (t) = JS (f) T (mv0, nr0) e2df = T (mv0, nr0) s (t) (33)
We will ask:

The demodulator therefore performs the following processing:

In the case of a stationary transfer function channel # ei #, we obviously find:
#m, n = pam, n (36)
In practice, the processing 107 is performed in digital form, according to the method illustrated in FIG. 11. The receiver operates in a similar manner to a receiver
OFDM / OQAM [13-16]. He performs the following treatments:
multiplication 111 of said received signal r (t) by the prototype function x (t)
112;
# fold # 113 of the modulo 2to filtered waveform;
- Application 114 of a Fourier transform (Fr);
correction 115 of phase e as a function of the estimate of channel 116,
comprising for example an estimate Pm.n of the amplitude response
and an estimate # m, n of the phase response of the transmission channel;
- correction 117 of the phase corresponding to the term im + n, the elements of
data being alternately in phase and in quadrature; selection 118 of the real part of the coceficient obtained a, @ corresponding
at the coefficient am, n emitted weighted by the amplitude response Pm, n of the
transmission channel.

 This algorithm therefore makes it possible to globally calculate all the coefficients of a given index n. The order of magnitude of the corresponding complexity is approximately twice that of the algorithm used for OFDM / QAM.

 The coefficients thus obtained are then de-interleaved 108, symmetrically with the interleaving implemented on transmission, and then decoded 109, advantageously according to a soft decision decoding technique, for example implementing an algorithm of the type of the algorithm of Viterbi. If the channel decoding takes into account the estimate of the response of the amplitude of the channel Pm n, the corresponding values are also deinterlaced 110. Furthermore, the deinterlacing is of course carried out before or after the mapping, depending on the moment when the interleaving was implemented on the program.

APPENDIX 1: REFERENCES [1] ML Doeltz, AND Heald and DL Martin, Binary data transmission techniques for
linear systems
Proceedings of the IVRE, pp. 656-661, May 1957.

[2] RR Mosier, A data transmission system using pulse phase modulation
IRE Conv. Rec. Ist Nat'l Conv Military Electronics (Washington DC, June 17-19, 1957) pp. 233-238.

[3] GA Franco and G. Lachs, An orthogonal coding technique for
communications 1961 IRE Internat'l Conv. Rec., Vol. 9, pp. 126-133.

[4] HF Harmuth, On the transmission of information by orthogonal time functions
AIEE Trans. (Communications and Electronics) vol. 79, pp. 248-255, July 1960.

[5] SB Weinstein and Paul M. Ebert, Data transmission by frequency-division
multiplexing using the discrete Fourier transform
IEEE Trans. Commun., Vol. COM-19, pp. 628-634, Oct. 1971.

[6] LJ Cimini, Analysis and simulation of a digital mobile channel using orthogonal
frequency division multiplexing,
IEEE Trans. Commun., Vol. COM-33, pp. 665-675, July 1985.

[7] EF Casas and C. Leung, OFDM for data communication over mobile FM radio
channels - Part I: Analysis and experimental results,
IEEE Trans. Commun., Vol. 39, pp. 783-793, May 1991.

[8] EF Casas and C. Leung, OFDM for data communication over mobile FM radio
channels - Part li: Performance improvement,
IEEE Trans. Commun., Vol. 40, pp. 680-683, April 1992.

[9] I. Daubechies, The transformed wavelet, time-frequency localization and signal
analysis,
IEEE Trans. Inform. Theory, vol. IT-36, pp. 961-1005, Sept. 1990.

[10] HE Jensen, T. Hoholdt, and J. Justesen, Double series representation of
bounded signals,
IEEE Trans. Inform. Theory, vol. IT-34, pp. 613-624, July 1988.

[1 1] RW Chang, Synthesis of band-limited orthogonal signals for multi-channel data
transmission,
Bell Syst. tech. J., vol. 45, pp. 1775-1796, Dec. 1966.

[12] BR Saltzberg, Performance of an efficient parallel data transmission system,
IEEE Trans. Common. Technol., Vol. COM-15, pp. 805-811, Dec. 1967.

[13] RW Chang, A theoretical study of orthogonal multiplexing data performance
transmission schema,
IEEE Trans. Common. Technol., Vol. COM-16, pp. 529-540, Aug. 1968.

[14] B. Hirosaki, An analysis of automatic equalizers for orthogonally multiplexed
QAM systems,
IEEE Trans. Commun., Vol. COM-28, pp. 73-83, Jan. 1980.

[15] B. Hirosaki, An orthogonally multiplexed QAM system using the Fourier discrete
transform,
IEEE Trans. Commun., Vol. COM-29, pp. 982-989, July. nineteen eighty one.

[16] B. Hirosaki, A maximum likehood receiver for an orthogonally mulitplexed QAM
system,
IEEE Journal on Selected Areas in Commun., Vol. SAC-22, pp. 757-764, Sept 1984.

[17] B. Hirosaki, S. Hasegawa, and A. Sabato, Advanced group-band modem using
orthogonally multiplexed QAM technique,
IEEE Trans. Commun., Vol. COM-34, pp. 587-592, June. 1986.

[18] John AC Bingham, Multicarrier Modulation for Data Transmission: An Idea
whose time has come,
IEEE Communications Magazine, pp. 5-14, May 1990.

[19] WOODWARD PM, Probability and information theory with application
Radar,
Pergamon Press,
London 1953.

[20] F. Amoroso and JA Kivett, Simplified MSK Technical Signaling,
IEEE Trans. Commun., Vol. COM-25, pp. 433441, April 1977.

[21] PA Bello, Characterization of Randomly Time-Variant Linear Channels,
IEEE Trans. Common. Systems, pp. 360-393, Dec. 1964.

[22] PM WOODWARD, Probability and information theory with application
Radar,
Pergamon Press,
London 1953.

[23] M. ALARD and R. LASSALLE Principles of channel modulation and coding in digital broadcasting to mobiles
EU Review R, no. 224, August 1987, pp. 168-190.

où s(t) et r(t) représentent respectivement les signaux émis et reçus # la réponse impulsionnelle en sortie (Output Delay Spread Function) h(t,T) définie par:

ANNEX 2 1. Channel modeling 1.1. General model
A dispersive channel can be considered as a linear system with a time-varying impulse response. There are two ways to define this impulse response. The conventions proposed in [21] will be largely inspired by:
the Input Delay Spread Function (Input Delay Spread Function)
g (t,) defined by:

where s (t) and r (t) respectively represent the transmitted and received signals # the Output Delay Spread Function h (t, T) defined by:

# la fonction d'étalement Doppler-délai V(v,l) (Doppler-Delay Spread
Function) caractérisée par:

Obviously, h (t,) = g (t + T) h (t, T) represents the impulse response of the channel at time t. With these conventions, we can define the following characteristic functions:
# the delay-Doppler spreading function U (, v) (Delay-Doppler Spread
Function) characterized by:

# Doppler-delay spread function V (v, l) (Doppler-Delay Spread
Function) characterized by:

We have simply: V (v, #) = ei2 # v # U (#, v) # the variable transfer function (Time-Variant Transfer Function) T (f, t) characterized by:

 We thus find the same equation as in the case of a stationary channel, the difference being simply that the transfer function becomes variable in time.

This transfer function T (f, t) is the two-dimensional Fourier transform of U (T, V), that is:

 In all cases, we will consider that U (T, V) has a bounded support, which makes it possible to represent the transfer function T (f, t) by an array of discrete values by virtue of the sampling theorem.

1.2. The static Doppler delay model
The delay-Doppler model is defined by the equation:

 This equation shows the channel as a sum of elementary channels characterized by amplitude, phase, time offset and frequency offset.

So it is legitimate to be interested in the behavior of different modulations in the presence of this type of channel, which we will call static Doppler delay.

ak étant des variables réelles, E étant un réseau bidimensionnel de densité 2 dans l'espace temps-fréquence, les fonctions xk(t) étant des translatées en temps et en fréquence d'une même fonction prototype x(t), et qui constituent une base hilbertienne de L2(R).

The equation of the channel is then written in the following simplified form:
r (t) = Aeios (t-z) e2 "2. OFDM performance in non-stationary channels 2.1 General case
Consider an OFDM multi-carrier modulation of any type (OFDM / QAM, OFDM / OQAM or OFDM / IOTA) characterized by the generic equation:

ak being real variables, E being a two-dimensional network of density 2 in the time-frequency space, the functions xk (t) being translated in time and in frequency of the same prototype function x (t), and constituting a Hilbert base of L2 (R).

It should be noted that no hypothesis is made on the structure of the network E. In the particular case of OFDM / QAM, this network is broken down into two sub-networks co-located with quadrature phases.

étant une phase estimée par le démodulateur et r(t) l'enveloppe complexe du signal reçu. On peut donc écrire:

The demodulation operation is written:

being a phase estimated by the demodulator and r (t) the complex envelope of the received signal. We can write:

Gold:

It follows that:

The optimal value of is that which maximizes the coefficient an, namely:

Although general, the equations above are hardly manipulable. They show, however, that the useful signal and the intersymbol appear as integrations of the function of ambiguity weighted by the function of delay spreading.
Doppler.

2.2. Static channel case
If one is interested in a channel of the static Doppler delay type, characterized by a phase 0, a delay X and an offset v (the amplitude A will be normalized to 1), the demodulation is carried out in a similar manner by introducing in the estimator a phase parameter. The result of this operation is written:

The demodulated signal is thus finally written:

The second term represents intersymbol interference (IES). Considering the data ak as independent random variables of variance a2, the variance I of the IES is written:

Now the coefficients ck are the coefficients of the decomposition of the function ei (# - #) e-2i @ v (# + #) xn (@ + T), of unit norm, on the Hilbert basis of the functions xk (t) . So we have:

In other words, the variance of the received signal is constant and is distributed between the "useful" signal cnan and the IES, of variance I = (1 - cn2) a2.The calculation of the coefficient cn gives:

Now the function of ambiguity of xn is written:
AXn (#, v) = e2i # (vn # - # nv) Ax (#, v)
Finally, we can write:

dans le cas général, qui représente la valeur quadratique moyenne de l'intersymbole normalisée par la valeur quadratique moyenne des données dans le cas d'une estimation de phase optimale.We will consider that the demodulation phase # is written in the form # opt + ##, where @pt is the demodulation phase that minimizes the IES, that is to say which maximizes cn, ie:
Seven = 8 + rvT + 2 # (# nv-vn #)
Then, the variance of the IES is simply written :: i = (1 - (e [Ax (T, v) e]) 2) a2
When the prototype function is even, (which corresponds to the case of the method of construction of Hilbert bases described in the main text), the function of ambiguity is real, and we have:
I = (1 - AX2 (T, V) COS2 i)) a2
This result is quite remarkable, since it demonstrates that the sensitivity to delay and Doppler of any multicarrier modulation depends only on the ambiguity function of its prototype function. Intersymbol function will be called later (by misuse of language, as a function of interference between symbols) the function

in the general case, which represents the mean square value of the intersymbol normalized by the mean squared value of the data in the case of an optimal phase estimate.

3. Comparative analysis of different types of OFDM 3.1. Theoretical limits
Here we are interested in the properties of the intersymbol function. It can be seen that the sensitivity of a multicarrier modulation is directly related to the behavior of the ambiguity function of the corresponding prototype function in the neighborhood of (0,0). The problem posed is quite analogous to the problems of uncertainty encountered in the radar field, and one can refer to the abundant literature on the subject (see for example [22]). Without loss of generality, one can choose the function x (t), by adequate temporal and frequency translation, so that its moments of order one are zero, ie:

Under these conditions, it is easy to check that partial derivatives of the first order cancel each other out:

We can characterize the behavior of the ambiguity function around (0,0) from second-order partial derivatives:

We will ask

Consider the Taylor-Young expansion of the ambiguity function in (0,0): Ax (d #, dv) = 1-2 # 2 (# t2dv2 + # f2d # 2) + dvd # + o (dv2 + d # 2)
We deduce Taylor-Young's development of the variance of the intersymbol
I = (l - (# e [Ax (#, v)]) 2 cos2 ##) # 2,
ie:: I (d #, dv, d #) = # 2 [4 # 2 (# t2dv @ + # f2d # 2) -2 dvd # + d # 2 + o (dv2 + d # 2 + d # 2)]
We deduce that the intersymbol function Is initially admits a tangent cone of equation:

The intersection of this cone with the plane z = 1 (maximum intersymbol) delimits an elliptic contour surface whose area # can be considered as a measure of sensitivity to delay and Doppler. When x is zero, this ellipse has as axis of symmetry the time and frequency axes, and extends by + 1 / 2Mf according to the time axis and # 1 / 2Xt according to the frequency axis. So we have: # = 1/4 ## t # f
By virtue of the Heisenberg inequality, # can not exceed unity. This result is generalized to the case where x is different from 0. Let us consider the function y (t) obtained by multiplying the function x (t) by a wobulation:

We can write:

 It is always possible to cancel lly by choosing ss appropriately. But the multiplication operation by a wobulation realizes a simple change of axes of the associated ambiguity function, with conservation of the areas. We deduce the parameter # is always between 0 and 1.

This result is extremely important, since it makes it possible to compare
performance of all MCMs in dispersive channels from a parameter
unique. It can thus be seen that these performances depend only on the concentration of
the associated prototype function. The optimum is reached virtually by the Gaussian,
but this optimum is inaccessible, since Gaussians do not allow
build a Hilbert base.

APPENDIX 3 1. Introduction
This appendix provides a prototype function construction method that satisfies the required orthogonality criteria. The method makes it possible to obtain an infinity of functions, among which a particular solution (called IOTA function) having the peculiarity of being identical to its Fourier transform.

2. Ambiguity function
This chapter recalls the main properties of the ambiguity function, and describes various operators acting on this function.

2.1 Reminders on the ambiguity function 2.1.1 Definitions
Let there be a function x (t) and its Fourier transform X (f). It can be associated with its temporal and frequency products defined respectively by:

2.1.2. Propriétés de symétrie de la fonction d'ambiguïté
Soit une fonction x(t).On notera respectivement par x et x les fonctions définies de la manière suivante:

<tb> Yx (t, T) <SEP> = <SEP> x (t + <SEP> r / 2) <SEP> x '(t- <SEP> r / 2) <SEP>
<tb> @x (f, v) = X (f + v / 2) X * (fv / 2)
<Tb>
The Wigner-Ville transform and the ambiguity function of x are then given by:

2.1.2. Symmetry properties of the ambiguity function
Let x be a function x (t). The functions defined by x and x respectively are defined as follows:

soit, en posant u = -t:

<tb><SEP> x @ (t) = x (-t)
<tb> x (t) <SEP> = <SEP> x * (- t) <SEP>
<Tb>
We then have relations:

either, by asking u = -t:

We conclude in particular that if a function x is even, that is x = x-, its ambiguity function is real. In addition, note the following relation:

ou encore :Ax (T, v) = Ax(v,#) 2.1.4.Fonction d'ambiguïté et translation temps fréquence
Considérons une fonction translatée d'une fonction prototype x(t) quelconque, soit:

By combining these two relations, we obtain:
A (r, v) = A (r, -v) 2.1.3. Ambiguity function and Fourier transform
The definition of the ambiguity function can be rewritten as follows:

Or again: Ax (T, v) = Ax (v, #) 2.1.4.Function of ambiguity and translation time frequency
Consider a translated function of any prototype function x (t), that is:

soit, en posant u = t - r,

2.2 Orthogonalité et fonction d'ambiguïté 2.2.1 Cas général
On considère deux fonctions translatées d'une même fonction x(t), soit:

The associated ambiguity function is written:

either, by putting u = t - r,

2.2 Orthogonality and ambiguity function 2.2.1 General case
We consider two functions translated from the same function x (t), that is:

soit, en posant u=t-(#k+#k')/2:

3. Bases hilbertiennes sur des réseaux orthogonaux 3.1. Principes généraux de construction
On considère un ensemble de fonctions {Xm,n} défini par:

The scalar product of these two functions is written:

or, by putting u = t - (# k + # k ') / 2:

3. Hilbert bases on orthogonal networks 3.1. General principles of construction
We consider a set of functions {Xm, n} defined by:

 We seek the conditions on x (t) for this set {Xm, n} to constitute a Hilbert basis of HR. We will impose that x (t) be an even function, whose ambiguity function Ax is therefore real.

The scalar product in R of Xm, n and of Xm ', n' n 'can be written

Note the congruence relation modulo 2:
(M-m ') + (n-n') + (m-m ') (n + n') # 1- (m-m '+ 1) (n-n' + 1)
Therefore, if (m, n) # (m ', n') modulo 2, the dot product is zero. The network {Xm, n} can therefore be broken down into four sub-networks characterized by: {m pair, n even}, {m pair, n odd}, {m odd n even}, {odd m, n odd}. The orthogonality between functions belonging to different subnets is therefore automatic, and does not depend on the properties of the prototype function, from the moment when it is even.

It remains to be ensured that the functions of the same sub-network are orthogonal to each other. All that is needed for this is that the ambiguity function Ax checks:
Ax (2n # 0.2mv0) = OV (m, n) $ (0.0)
So we see that the problem of the construction of HR Hilbert bases
on an orthogonal network of density 2 is reduced to that of the construction of an even prototype function whose ambiguity function vanishes on a network of density 1/2.

3.2. Orthogonalization methods 3.2.1. Temporal onhogonalisation
Definition:
Let there be a function x (t) of Fourier transform X (f). O @ is the temporal orthogonalization operator that associates with x (t) a function y (t) defined by its Fourier transform Y (f):

soit, par transformée de Fourier inverse:

By construction, we have:

or, by inverse Fourier transform:

is still
Ay (2n # 0.0) = 0 # n # 0 and Ay (0,0) = 1
The orthogonalization is thus well realized on the temporal axis. Note further that this operator normalizes y.

Let x be a Gaussian function and y = Otx. Consider the expression:

Since X is a Gaussian, we can write:
X (f + mvO) X (f - mvO) = cm | X (f) | 2
where cm is a constant. We deduce that: #y (f, 2mv0) = cm # y (f, 0)
By inverse Fourier transform, we obtain: Ay (#, 2mv0) = cmAy (#, 0)
Therefore: # m, # n ## 0 Ay (2n # 0,2mv0) = 0
The time orthogonalization operator Ot orthogonalizes the entire network, with the exception of the frequency axis.

Theorem 1
Let x be a Gaussian function and y = Otx, then: # m, # n ## 0 Ay (2n # 0,2mv0) = 0 3.2.2. Frequency orthogonalization
Definition
Let be a function x (t). Of is the frequency orthogonalization operator that associates with x (t) a function y (t) defined by:

soit, par transformée de Fourier:

By construction, we have:

either, by Fourier transform:

is still
Ay (0, 2mv0) = 0 # m # 0 and Ay (0,0) = l
Orthogonalization is therefore well realized on the frequency axis. Note further that this operator normalizes y.

Let x be a Gaussian function and z = Ofy, with y = Otx. Consider the expression:

We can therefore write: &gamma; z (t, 2n # 0) - &gamma; y (t, 2n # 0) P (t) where P (t) is a periodic function of period 0, which admits a series development of Fourier of the type

By Fourier transform, we obtain:

Gold

In addition, by construction,
# m # 0, Az (0, 2mv0) = 0
So we have finally: # (m, n) # (0,0), Az (2n # 0,2mv0) = 0
Thus, the ambiguity function of z vanishes outside (0.0) for all the multiples of 210 and 2v0, ie a density network 1/2.

Theorem 2
Let x be a Gaussian function and z = OfOtX, then: # (m, n) # (0,0), Az (2n # 0,2mv0) = 0 3.3. The orthogonalization operator O
In view of the foregoing; it is clear that there is a time-frequency scale which symmetrizes the writing of the equations: it suffices for that to choose # 0 = v0 = 1 / W. We will renormalize the scales accordingly, without affecting the generality of the demonstrations. .

3.3.1. Definition
We call O the orthogonalization operator that associates with a function x the function y defined by:

 In addition, the operator of the Fourier transform will subsequently be called F.

3.3.2. Idempotence of the operator O
Let z = Oy and y = Ox. We can write:

 We thus have OOx = Ox, which shows the idempotence of the operator O. In the same way, the dual operator F-1OF is also idempotent, since F-1OFF-1OF = F-1OOF = F-1OF.

3.3.3. Lemma 1
Let P be a periodic function of period 1/4 and D a distribution of the form:

Let x be any function:

Lemma 1
Let P be a periodic function of period 1 / and D a distribution of the form

Let x be any function.
D * (P x) = P (D * x)
Lemma 2
Let the function y be defined by Ya = D * xa, with x &alpha; = (2 &alpha;) 1/4 e ~, and D being a distribution of the form

We can write:

Consider the sum:

puis, en réorganisant les indices et en redéfinissant k comme k + k' + k":

In other words, by applying the result given in appendix (≈4):

then, by rearranging the indices and redefining k as k + k '+ k ":

avec

We can write:

with

We can easily estimate the coefficient c by rewriting the previous relation in the form:

Either, by Fourier transform:

In particular, we deduce: || y &alpha; || 2 = Ay &alpha; (0,0) = cAx &alpha; (0,0) = c || x &alpha; || 2
So we finally have ::

Lemma 2
Let Ya be the function Ya = D * x &alpha;, with x &alpha; = (2 &alpha;) 1/4 # - # &alpha; u2, and D being a distribution of the form

3.3.5. Commutativité des opérateurs O et F-1OF
Nous allons maintenant démontrer que les opérateurs O et F-1OF commutent lorsqu'ils sont appliqués à une gaussienne. Soit x&alpha;=(2&alpha;)1/4e-#&alpha;u2. We can write:

3.3.5. Commutativity of operators O and F-1OF
We will now show that the operators O and F-1OF switch when they are applied to a Gaussian. Let x &alpha; = (2 &alpha;) 1 / 4e - # &alpha; u2.

et sa transformée de Fourier D, par:

Then Fxa = x1 / &alpha; and O xa = PaXa
P &alpha; being defined by the relation:

and its Fourier transform D, by:

Xa et Ya étant de norme unité, on peut écrire, en application du lemme 2:

Let Ya = F-1OFx &alpha; and za = Oy &alpha;. We can write: y &alpha; = F-1OFx &alpha; = F-1Ox1 / &alpha; = F-1 (P1 / &alpha; x1 / &alpha;) = D1 / &alpha; * x &alpha;

Xa and Ya being of norm unit, one can write, in application of lemma 2:

x1/&alpha; @ &alpha; et w" a étant de norme unité, on a, en application du lemme 2:

In the same way, we define: w1 / &alpha; = FOX &alpha; = F (P &alpha; x &alpha;) = D &alpha; * x1 / &alpha;
We can write:

x1 / &alpha; @ &alpha; and w "a being of unit norm, we have, in application of lemma 2:

Either by inverse Fourier transform: F-1OFOx &alpha; = F-1Ow1 / &alpha; = D1 / &alpha; * (P &alpha; x &alpha;)
Now, by the application of lemma 1:
Dua * (Paxa) = Pa (D1. &Alpha; * xa)
We deduce that: OF-1OFx &alpha; -F-1OFOx &alpha;
Theorem 3
For any Gaussian function x, the operators O and F-tOF switch, that is:
OF-F-1OFx = 1OFOx
Corollary 1
Let z &alpha; = OF-1OFx &alpha;, with x &alpha; = (2 &alpha;) 1/4 e - # &alpha; u2, then Fz &alpha; = z1 / &alpha;
Demonstration::
Fz &alpha; = FF-1OFOx &alpha; = OF-1Ox &alpha; = OF-1OFx1 / &alpha; = z1 / &alpha;
Particular case remarkable
FZL-zl
This particular function gives a perfect symmetry to the time axes and
frequency, and thus constitutes the prototype function of the IOTA transform (Isotropic
Orthogonal Transform Algorithm). Note this particular function 9.

Corollary 2
Let x be a Gaussian function and z = OF-1OFx, then Oz = z
Demonstration:
Oz = OOF-1OFx = OF-1OFx = z
Corollary 3
Let x be a Gaussian function and z = OF-1OFx, then F-1OFz = z
Demonstration:
F-1OFz = F-1OFF-1OFOx = F-1OOFOx = F-1OFOX = z 3.3.6. Function of ambiguity of the functions z
Consider Theorem 2, with the normalization # 0 = V0 = l / W2. So:
Of-OetOt-1of F =
Therefore, Theorem 2 can be rewritten:
Theorem 4
Let x be a Gaussian function and z = F-1OFOx, then: # (m, n) # (0,0), Az (n # 2, m # 2) = 0 4. Appendix
Let be a normalized Gaussian function Xa defined by: x &alpha; (u) = (2 &alpha;) 1 / 4e - # &alpha; u2
The product x &alpha; (ua) x &alpha; (ub) can thus be written:

But we have the identity:

a + b <SEP> ab
<tb> (ua) 2+ (ub) 2 = 2 ## u- # 2 + ## 2 #
<tb> 2 <SEP> 2
<Tb>
Finally, we can write:

Claims (8)

CLAIMS 1. Multicarrier signal intended to be transmitted to digital receivers, in particular in a non-stationary transmission channel, corresponding to the multiplexing in frequency of several elementary carriers each corresponding to a series of symbols, two consecutive symbols being separated by a time symbol 0, characterized firstly by the fact that the spacing v0 between two neighboring carriers is equal to half the inverse of the symbol time 0, and secondly that each carrier undergoes a shaping filtering its spectrum having a bandwidth strictly greater than twice said carrier spacing v0, and chosen so that each symbol is highly concentrated in the time domain and the frequency domain. 2. Signal according to claim 1, characterized in that its complex envelope meets the following equation:

 or:  a.n is a real coefficient, chosen from a predetermined modulation alphabet;  m is an integer representing the frequency dimension;  n is an integer representing the time dimension;  t represents the time;  x ", n (t) is a basic function, translated in the time-frequency space of a  same prototype function x (t) pair taking real or complex values, either ::

 where (p is an arbitrary phase parameter,  the Fourier transform X (f) of the function x (t) having a support extending to  beyond the interval [-v0, v0], and where said basic functions {xmn) are orthogonal to each other, the real part of the scalar product of two different basic functions being zero. 3. Signal according to claim 2, characterized in that said prototype function x (t) is an even function, zero outside the interval [-T0, T0], and verifying the relation:

<tb> <SEP> x (t) = 0 <SEP> if | t | ## 0 <tb> # <tb> <SEP> | x (t) | 2+ | x (t- # 0) | 2 <SEP> <SEP> = l / # 0 <SEP> if <SEP> 0 <SEP> #t <# 0 <tb> 4. Signal according to claim 3, characterized in that said prototype function x (t) is defined by:

If t S  elsewhere 5. Signal according to claim 2, characterized in that said prototype function x (t) is characterized by the equation:

 the function y (t) being defined by its Fourier transform Y (f):

 where G (f) is a normalized Gaussian function of the type: G (f) = (2a) "4e-t, a being a strictly positive real parameter. 6. Signal according to claim 5, characterized in that the parameter a is equal to unity. 7. A method for transmitting a digital signal, in particular in a non-stationary transmission channel, characterized in that it comprises the following steps:  - channel coding of a digital signal to be transmitted, delivering coefficients  actual numerals am, n selected in a predetermined alphabet;  constructing a signal s (t) corresponding to the following equation:

 or: . m is an integer representing the frequency dimension; . n is an integer representing the time dimension; . t represents the time; . xm, n (t) is a basic function, translated in the time-frequency space of the same prototype function x (t) pair taking real or complex values, ie:

 where (p and an arbitrary phase parameter,  the Fourier transform X (f) of the function x (t) having a support  extending beyond the interval [-v0, v0], said basic functions [xm, n] being orthogonal to each other, the part  real scalar product of two different basic functions being  nothing;  transmitting a signal having for complex envelope said signal s (t) towards  least one receiver. 8. Method according to claim 7, characterized in that it comprises a frequency interleaving step and / or time, applied to the bits forming said digital signal to be transmitted or digital coefficients amn from the channel coding. 9. Method for receiving a signal according to any one of claims 1 to 7, characterized in that it comprises the following steps:  receiving a signal having a complex signal envelope r (t);  - estimation of the response of the transmission channel, including an estimate of  the response in the 6m phase, n and the amplitude response Pm n;  demodulating said signal r (t), comprising the following steps:  multiplying said signal r (t) by the prototype function x (t);  folding of the waveform obtained modulo 210;  - application of a Fourier transform (Wr);  - correction of the Omn phase induced by the transmission channel;  correction of the phase corresponding to the term im + n;  selection of the real part of the coeeficient obtained am, n corresponding to the coefficient am n emitted weighted by the amplitude response # m, n of the channel of  transmission. 10. Method according to claim 9, characterized in that it comprises a step of deinterlacing in frequency and / or in time of said real digital coefficients am.n  and, possibly, corresponding values # m, n of the amplitude response of the channel,  said deinterleaving being symmetrical to an interlacing implemented on transmission,  and / or a weighted decision decoding step adapted to the channel coding implemented at  transmission.  11. A method of constructing a prototype function x (t) for a signal according to any one of claims 2 to 6, characterized in that it comprises the following steps  - selection of a Gaussian function G (f) normalized of the type: G (f) = (2 &alpha;) 1 / 4e - # &alpha; f  determining said prototype function x (t) such that:

 the function y (t) being defined by its Fourier transform Y (f):